This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate? Excessive Violence Sexual Content Political / Social
Email Address:
Article Id: WHEBN0000229939 Reproduction Date:
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator
and the integration operator J. (Usually J is used instead of I to avoid confusion with other I-like glyphs and identities.)
In this context, the term powers refers to iterative application of a linear operator acting on a function, in some analogy to function composition acting on a variable, e.g., f ^{2}(x) = f(f(x)). For example, one may ask the question of meaningfully interpreting
as an analog of the functional square root for the differentiation operator (an operator half iterated), i.e., an expression for some linear operator that when applied twice to any function will have the same effect as differentiation.
More generally, one can look at the question of defining
for real-number values of a in such a way that when a takes an integer value n, the usual power of n-fold differentiation is recovered for n > 0, and the −nth power of J when n < 0.
The motivation behind this extension to the differential operator is that the semigroup of powers D^{a} will form a continuous semigroup with parameter a, inside which the original discrete semigroup of D^{n} for integer n can be recovered as a subgroup. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent a, since it need not be rational; the use of the term fractional calculus is merely conventional.
Fractional differential equations (also known as extraordinary differential equations) are a generalization of differential equations through the application of fractional calculus.
An important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integer cases we cannot say that the fractional derivative at x of a function f depends only on values of f very near x, in the way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. To use a metaphor, the fractional derivative requires some peripheral vision.
As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832. The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.^{[1]}
A fairly natural question to ask is whether there exists an operator H, or half-derivative, such that
It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that
or to put it another way, the definition of d^{n}y/dx^{n} can be extended to all real values of n.
Let f(x) be a function defined for x > 0. Form the definite integral from 0 to x. Call this
Repeating this process gives
and this can be extended arbitrarily.
The Cauchy formula for repeated integration, namely
leads in a straightforward way to a generalization for real n.
Using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for fractional applications of the integral operator.
This is in fact a well-defined operator.
It is straightforward to show that the J operator satisfies
This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative nor additive in general.
Let us assume that f(x) is a monomial of the form
The first derivative is as usual
Repeating this gives the more general result that
Which, after replacing the factorials with the gamma function, leads us to
For k=1 and \textstyle a=\frac{1}{2}, we obtain the half-derivative of the function x as
Repeating this process yields
which is indeed the expected result of
For negative integer power k, the gamma function is undefined and we have to use the following relation:^{[2]}
This extension of the above differential operator need not be constrained only to real powers. For example, the (1 + i)th derivative of the (1 − i)th derivative yields the 2nd derivative. Also notice that setting negative values for a yields integrals.
For a general function f(x) and 0 < α < 1, the complete fractional derivative is
For arbitrary α, since the gamma function is undefined for arguments whose real part is a negative integer, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,
We can also come at the question via the Laplace transform. Noting that
and
etc., we assert
For example
as expected. Indeed, given the convolution rule
and shorthanding p(x) = x^{α−1} for clarity, we find that
which is what Cauchy gave us above.
Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations.
The classical form of fractional calculus is given by the Riemann–Liouville integral, which is essentially what has been described above. The theory for periodic functions (therefore including the 'boundary condition' of repeating after a period) is the Weyl integral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to 0).
By contrast the Grünwald–Letnikov derivative starts with the derivative instead of the integral.
The Hadamard fractional integral is introduced by J. Hadamard ^{[3]} and is given by the following formula,
Not like classical Newtonian derivatives, a fractional derivative is defined via a fractional integral.
The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing n-th order derivative over the integral of order (n − α), the α order derivative is obtained. It is important to remark that n is the nearest integer bigger than α.
There is another option for computing fractional derivatives; the Caputo fractional derivative. It was introduced by M. Caputo in his 1967 paper.^{[4]} In contrast to the Riemann Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows.
The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940).^{[5]} and Hermann Kober (1940)^{[6]} and is given by
which generalizes the Riemann fractional integral and the Weyl integral. A recent generalization is the following, which generalizes the Riemann-Liouville fractional integral and the Hadamard fractional integral. It is given by,^{[7]}
In the context of functional analysis, functions f(D) more general than powers are studied in the functional calculus of spectral theory. The theory of pseudo-differential operators also allows one to consider powers of D. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potentials. So there are a number of contemporary theories available, within which fractional calculus can be discussed. See also Erdélyi–Kober operator, important in special function theory (Kober 1940), (Erdélyi 1950–51).
As described by Wheatcraft and Meerschaert (2008),^{[8]} a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:
This equation has been shown useful for modeling contaminant flow in heterogenous porous media.^{[9]}^{[10]}^{[11]}
Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.^{[12]}^{[13]} The time derivative term is corresponding to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as
A simple extension of fractional derivative is the variable-order fractional derivative, the α, β are changed into α(x, t), β(x, t). Its applications in anomalous diffusion modeling can be found in reference.^{[14]}
Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.^{[15]}
The propagation of acoustical waves in complex media, e.g. biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
See also ^{[16]} and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in ^{[17]} and in the survey paper,^{[18]} as well as the acoustic attenuation article. See ^{[19]} for a recent paper which compares fractional wave equations which model power-law attenuation.
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics discovered by Nick Laskin,^{[20]} has the following form:^{[21]}
where the solution of the equation is the wavefunction ψ(r, t) - the quantum mechanical probability amplitude for the particle to have a given position vector r at any given time t, and ħ is the reduced Planck constant. The potential energy function V(r, t) depends on the system.
Further, Δ = ∂^{2}/∂r^{2} is the Laplace operator, and D_{α} is a scale constant with physical dimension [D_{α}] = erg^{1 − α}·cm^{α}·sec^{−α}, (at α = 2, D_{2} = 1/2m for a particle of mass m), and the operator (−ħ^{2}Δ)^{α/2} is the 3-dimensional fractional quantum Riesz derivative defined by
The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2.
Isaac Newton, Gottfried Wilhelm Leibniz, Integral, Electromagnetism, Mathematics
Isaac Newton, Integral, Calculus, Differentiation rules, Gradient
Integral, Dot product, Directional derivative, Mathematics, Vector field
Linear algebra, Banach space, Integral, Hilbert space, Game theory
Calculus, Integral, Rolle's theorem, Gradient, Continuous function
Nyquist plot, Fractional calculus, Bode plot, Overshoot (signal), Step response
Kerala, Government of India, India, Canada, Kottayam
Arima, Statistics, Parameter, Long-range dependency, Fractional Brownian motion